Estimating the relative importance of vertices and edges is a fundamental issue in the analysis of complex networks, and has found vast applications in various aspects, such as social networks, power grids, and biological networks. Most previous work of centralities focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called theta-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter theta. We define two equivalent measures for theta-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality.

Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the theta-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its epsilon-approximation for all the m edges in nearly linear time in m. The proposed theta-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the $theta$-Kirchhoff edge centrality, we present a theta-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute epsilon-approximate Kirchhoff vertex centrality for all the n vertices in nearly linear time in m.